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Uniform convergence rate for Birkhoff means of certain uniquely ergodic toral maps

Part of: Harmonic analysis in one variable Ergodic theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  06 October 2020

SILVIUS KLEIN Open the ORCID record for SILVIUS KLEIN [Opens in a new window] ,
XIAO-CHUAN LIU  and
ALINE MELO
SILVIUS KLEIN*
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Brazil (e-mail: [email protected])
XIAO-CHUAN LIU
Affiliation:
Instituto de Matemática e Estatística da Universidade de São Paulo, R. do Matão, 1010 - Vila Universitaria, São Paulo, Brazil (e-mail: [email protected])
ALINE MELO
Affiliation:
Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Brazil (e-mail: [email protected])
*
e-mail: [email protected]
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Abstract

We obtain estimates on the uniform convergence rate of the Birkhoff average of a continuous observable over torus translations and affine skew product toral transformations. The convergence rate depends explicitly on the modulus of continuity of the observable and on the arithmetic properties of the frequency defining the transformation. Furthermore, we show that for the one-dimensional torus translation, these estimates are nearly optimal.

Keywords

classical ergodic theoryconvergence of Birkhoff meansDenjoy–Koksma inequalityDiophantine conditionFejér and Jackson kernels

MSC classification

Primary: 37A45: Relations with number theory and harmonic analysis 37A30: Ergodic theorems, spectral theory, Markov operators
Secondary: 42A10: Trigonometric approximation 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series 11K06: General theory of distribution modulo $1$
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3363 - 3388
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Avila, A., Fayad, B. and Krikorian, R.. A KAM scheme for SL(2,  $\mathbb{R}$ ) cocycles with Liouvillean frequencies. Geom. Funct. Anal. 21(5) (2011), 1001–1019.CrossRefGoogle Scholar
Cai, A., Duarte, P. and Klein, S.. Large deviations for mixed random-quasiperiodic cocycles. In preparation.Google Scholar
Damanik, D.. Schrödinger operators with dynamically defined potentials. Ergod. Th. & Dynam. Sys. 37(6) (2017), 1681–1764.CrossRefGoogle Scholar
Dolgopyat, D. and Fayad, B.. Limit theorems for toral translations. Hyperbolic Dynamics, Fluctuations and Large Deviations (Proceedings of Symposia in Pure Mathematics, 89). American Mathematical Society, Providence, RI, 2015, pp. 227–277.CrossRefGoogle Scholar
Duarte, P. and Klein, S.. Continuity of the Lyapunov Exponents of Linear Cocycles . Publicações Matemáticas do IMPA [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2017.Google Scholar
Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.CrossRefGoogle Scholar
Grafakos, L.. Classical Fourier Analysis ( Graduate Texts in Mathematics , 249), 3rd edn. Springer, New York, NY, 2014.Google Scholar
Herman, M.-R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. Inst. Hautes Études Sci. 49 (1979), 5–233.CrossRefGoogle Scholar
Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1997.Google Scholar
Katok, A. B.. Combinatorial Constructions in Ergodic Theory and Dynamics. Vol. 30. American Mathematical Society, Providence, RI, 2003.Google Scholar
Katznelson, Y.. An Introduction to Harmonic Analysis, 3rd edn. Cambridge University Press, Cambridge, 2004. MR 2039503.CrossRefGoogle Scholar
Lang, S.. Introduction to Diophantine Approximations, 2nd edn. Springer, New York, NY, 1995. MR 1348400.CrossRefGoogle Scholar
Hugh, L.. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis ( CBMS Regional Conference Series in Mathematics , 84). Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994.Google Scholar
Queffélec, H. and Queffélec, M.. Diophantine Approximation and Dirichlet Series ( Harish-Chandra Research Institute Lecture Notes , 2). Hindustan Book Agency, New Delhi, 2013.CrossRefGoogle Scholar
Shadrin, A.. Lecture notes for the course on approximation theory at the University of Cambridge, http://www.damtp.cam.ac.uk/user/na/PartIIIat/at.html.Google Scholar
Stein, E. M. and Shakarchi, R.. Fourier Analysis ( Princeton Lectures in Analysis , 1). Princeton University Press, Princeton, NJ, 2003.Google Scholar
Yoccoz, J.-C.. Sur la disparition de propriétés de type Denjoy-Koksma en dimension 2. C. R. Acad. Sci. Paris Sér. A-B 291(13) (1980), A655–A658.Google Scholar
Yoccoz, J.-C.. Centralisateurs et conjugaison différentiable des difféomorphismes du cercle. Astérisque 231 (1995), 89–242.Google Scholar
Zhou, Q. and Wang, J.. Reducibility results for quasiperiodic cocycles with Liouvillean frequency. J. Dynam. Differential Equations 24(1) (2012), 61–83.CrossRefGoogle Scholar